Exponentials & Natural Logarithms Recognise or sketch an exponential y = Ae^(kx) and a logarithmic graph y = lnx Recall the main features of the exponential functions Recall the main features of the logarithmic function Explain the inverse relationship between e and ln

Exponential Recap and new features on GDC In pairs, go to the GRAPH function of your GDC. One of you should plot y = 2x and the other should plot y = 3x Everyone should plot a second graph as instructed on the next slide…

Exponential Recap and new features on GDC To plot the second graph Press OPTN Press CALC (F2) Press d/dx (F1) Press Y (F1) then 1. This is saying differentiate equation Y1. If you used another space on your graph like Y2, use Y2 instead. Plot the graph. You might have to change the scales, so F3 and change the min x to -2, max x to 2, and the scale to 0.5. Redraw.

Exponential Recap and new features on GDC In your pairs, discuss the two graphs you’ve just plotted. Come up with 3 different observations (they could be similarities or differences).

Observations y = 2x y = 3x dy dx dy dx There should be a value where y = Ax is the same as it’s derivative.

Let’s find that value On your GDC, press MENU and go to the TABLE function. Hopefully, you should still have your two equations. First, let’s set out table. Press F5 and change your settings to mine.

Let’s find that value dy dx y = 2x If you draw your table, you should see two values for y. One is for your exponential. One is for the derivative. Next, press EXIT. Using Trial & improvement, replace the 2 or 3 in or until the two y values are as close as possible (when is the derivative the same as the original?) You’ve got 5 mins…who can get the most significant figures??? y = 2x y = 3x

e This value you came up with is very special! We call it e. It’s just like π in that it is an irrational number. The value of e is 2.7182818284… It’s special in that e appears in real life, anything that has exponential growth or decay, is related to e in someway. A.k.a. rabbit population, interest rates, radioactive decay, and cooling coffee

y = ex graphs In your pairs, come up with three properties of the y =ex graph. Horizontal asymptote as the graph approaches negative infinity at y = 0. Y-intercept at (0, 1). Increases exponentially from left to right.

So how do we undo a y = ex function? We change it to a log form Here we have something interesting, We call this natural log, , and it stands for logarithmus naturalis. Recall the rule: , so if we take the which is normally written as , we get 1. Note: e and ln are inverses.

In your pairs, jot down at least 3 log rules using ln instead. Natural Logs Natural logs following the same rules as common logs. In your pairs, jot down at least 3 log rules using ln instead.

Natural logs

In your pairs, come up with 3 observations and the two graphs. Time to plot some more graphs. Go back to the GRAPH function, and plot y = ex and y = lnx. In your pairs, come up with 3 observations and the two graphs.

Natural Log graph Reflection in the line y = x. The are inverses of each other. y = ex has a y-intercept of (0, 1) whereas y = lnx has an x-intercept of (1, 0). y = ex has a horizontal asymptote at y = 0, and y = lnx has a vertical asymptote at x = 0.

Inverse relationship On your RUN MAT feature on your GDC, try the following Apply the rules above to help you solve the following:

Exponentials & Natural Logarithms Fill in your self-assessment (smiley face) sheet Recognise or sketch an exponential y = Ae^(kx) and a logarithmic graph y = lnx Recall the main features of the exponential functions Recall the main features of the logarithmic function Explain the inverse relationship between e and ln

Homework Check weekly homework email! ALL independent study will be checked during the first lesson each week.